Relative liveness: from intuition to automated verification

We point out deeciencies of previous treatments of liveness. We deene a new liveness condition in two forms: one based on nite trace theory, and the other on automata. We prove the equivalence of these two deenitions. We also introduce a safety condition and provide modular and hierarchical veriication theorems for both safety and liveness. Finally, we present a veriication algorithm for liveness.